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The author has developed a theory on optimal
trajectories for air vehicles with variable wing areas and with conventional
wings. He applied a new theory of
singular optimal solutions and obtained in many cases the optimal flight. The
wing drag of a variable area wing does not depend on air speed and air
density. At first glance the results may
seem strange, however, this is the case and this chapter will show how the new
theory may be used. The equations that
follow enable computations of the optimal control and optimal trajectories of
subsonic aircraft with pistons, jets, and rocket engines, supersonic aircraft,
winged bombs with and without engines, hypersonic warheads, and missiles with
The main idea
of the research is to use the vehicle’s kinetic energy to increase the range of
missiles and projectiles.
shows that the range of a ballistic warhead can be increased 3–4 times if an
optimal wing is added to it, especially a wing with variable area. If we do not
need increased range, the head mass of rockets can be increased. The range of
large gun shells can also be increased 3–9 times. The range of an aircraft may
be improved by 3–15% or more.
The results can be used for the design of aircraft,
space ship, head of rockets, missiles, flying apparatus and shells for large